Why 1.22 and not another number?

It is the ratio of the first zero of J₁ to π, (3.8317…)/π ≈ 1.22, for a circular aperture. Slits use different geometry (sinc, not Airy).

Does the two-source mode show coherent interference?

No — intensities add (incoherent point sources). This matches the usual Rayleigh discussion for unresolved stars.

Airy Disk & Rayleigh Limit

Circular aperture Fraunhofer pattern; first dark ring; two-point resolution.

About this model

For a circular aperture of diameter D, the Fraunhofer diffraction pattern at screen distance L has radial intensity I(r) ∝ [2J₁(v)/v]² with v = π D r/(λ L). The first dark ring occurs at the first zero of J₁, v ≈ 3.8317, giving r₁ ≈ 1.22 λL/D. Rayleigh’s resolution criterion for two incoherent point sources of equal brightness uses the angular separation θ ≈ 1.22 λ/D (central maximum of one on the first minimum of the other).

Who it's for: After slit diffraction; connects telescope/microscope resolution to aperture size and wavelength.

Key terms

Airy disk
Bessel function
Fraunhofer diffraction
Rayleigh criterion
angular resolution

How it works

Fraunhofer diffraction by a circular aperture gives the Airy pattern: a bright central disk and concentric rings. Intensity I/I₀ = [2J₁(v)/v]² with v = π D r /(λ L) on a screen at distance L. The first dark ring sets the size of the Airy disk. Rayleigh’s criterion quotes a minimum resolvable angular separation θ ≈ 1.22 λ/D between two incoherent point sources of equal brightness. Toggle two sources and scan separation in units of r₁.

Key equations

v = π D r / (λ L) · I ∝ [2J₁(v)/v]²

θ_Rayleigh ≈ 1.22 λ/D · r₁ ≈ 1.22 λL/D

Frequently asked questions

Why 1.22 and not another number?: It is the ratio of the first zero of J₁ to π, (3.8317…)/π ≈ 1.22, for a circular aperture. Slits use different geometry (sinc, not Airy).
Does the two-source mode show coherent interference?: No — intensities add (incoherent point sources). This matches the usual Rayleigh discussion for unresolved stars.

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